In this subsection, we briefly compare the cost of computing a full singular value decomposition with that of computing an approximate decomposition. This is a vast topic in numerical analysis and we refer the reader to  [1]}, [2]}, [3]}, [4]}, [5]} for implementation and complexity details. We assume that the matrix we wish to factor is dense with dimension \(m\times n\) . Broadly speaking, there are two approaches to computing an SVD: direct methods based on a QR factorization, and iterative methods such as the Krylov family of algorithms. For computing an approximate SVD, one can use the the truncated method (REF ) which involves computing a full SVD (which may not be practical for large \(m,n\) ), a rank-revealining QR decomposition can be used, and finally randomized methods can be used.
